Related papers: Conditions de Kan sur les nerfs des $\omega$-cat\'…

We show that C if is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. The strict model structure is the starting point for many homotopy…

In this paper we show an explicit relation between chains of torsion classes and weak stability conditions over an abelian category. In particular, up to a natural equivalence, they coincide. We investigate topological properties of the…

Many definitions of weak n-category have been proposed. It has been widely observed that each of these definitions is of one of two types: algebraic definitions, in which composites and coherence cells are explicitly specified, and…

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations for which the…

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose…

Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a…

We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before…

We study weakly invertible cells in weak $\omega$-categories in the sense of Batanin-Leinster, adopting the coinductive definition of weak invertibility. We show that weakly invertible cells in a weak $\omega$-category are closed under…

Given a locally finite cover of a simplicial complex by subcomplexes, Bj\"orner's version of the Nerve Theorem provides conditions under which the homotopy groups of the nerve agree with those of the original complex through a range of…

We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan…

We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $\omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that…

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric…

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (infinity,1)-categories because they provide a means for comparing two models of their respective homotopy…

Modular operads are an extension of operads. In the same way that operads, as dendroidal sets, can be considered as presheaves over the category of trees, so can modular operads be considered as presheaves over a category of graphs. This…

For a class of affine algebraic groups $\mathcal C$ over a field, we define the notions of $\mathcal C$-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient…

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal…

A class of partially wrapped Fukaya categories in $T^* N$ are proven to be well defined and then studied. In the case of $N$ diffeomorphic to $\mathbb{R}^m \times \mathbb{T}^n$, it is shown that these categories provide homological mirrors…

While chain complexes are equipped with a differential $d$ satisfying $d^2 = 0$, their generalizations called $N$-complexes have a differential $d$ satisfying $d^N = 0$. In this paper we show that the lax nerve of the category of chain…

For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of…

In this article, we prove that a complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded…